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jstevh@xxxxxxx wrote:
> Some recent threads simply explaining yet another way to see the
> coverage problem of the ring of algebraic integers have floundered on
> the issue of convergence and what it means with rings like the ring of
> algebraic integers, where I introduce the concept of an exclusionary
> ring.
>
> The concepts are simple, luckily, as I can mostly use something basic:
>
> S = 1 + x + x^2 + x^3 +...
>
> where the issue is convergence, so that if S converges in whatever ring
> you're in--notice none given yet--you can go to
>
> S = 1 + x*S
>
> and solve to get
>
> S = 1/(1-x)
>
> where to get convergence in the ring of complex numbers, which of
> course is also the field of complex numbers, let's choose x = sqrt(5) -
> 2, considering only the positive solution, then in complex numbers, I
> have
>
> S = 1/(1 - sqrt(5))
>
> which is a complex number, and it's also an algebraic number, but it's
> NOT an algebraic integer.
>
> But if I go back to
>
> S = 1 + x + x^2 + x^3 +...
>
> and plug in x=sqrt(5) - 2, I'll have
>
> S = 1 + (sqrt(5) - 2) + (sqrt(5) - 2)^2 + (sqrt(5) - 2)^3 +...
>
> and if I stop at some point like just look at
>
> 1 + (sqrt(5) - 2) + (sqrt(5) - 2)^2
>
> I have an algebraic integer, and in fact, I can have an arbitrarily
> large number of terms added together, and stop--ever closer to
> 1/(1-sqrt(5)) and STILL have an algebraic integer.
>
> BUT in the ring of algebraic integers, you cannot reach 1/(1-sqrt(5))
> because it is NOT an algebraic integer!
>
> Yet you can approach it out to infinity, so you have an asymptotic
> approach to that value.
>
> A corollary to this is 1/x where you can let x go out to infinity and
> that approaches 0 but never reaches it, or you can let x approach 0,
> but never reach it, but the difference with the infinite sum example is
> that the asymptotic nature is created by the exclusionary nature of the
> ring of algebraici integers!!!
>
> That is, the reason you can never reach 1/(1-sqrt(5)) with the series
> is that 1/(1-sqrt(5)) is NOT the root of any monic polynomial with
> integer coefficients, which is the rule that defines algebraic integers
> and excludes that value!
>
> So the mathematics holds on a definition, as logically, if
> 1/(1-sqrt(5)) is not an algebraic integer, then there is no way to
> reach that value in the ring, so
>
> S = 1 + (sqrt(5) - 2) + (sqrt(5) - 2)^2 + (sqrt(5) - 2)^3 +...
>
> approaches it asymptotically, in the ring of algebraic integers.
>
> But now you have a problem with
>
> S = 1 + x*S
>
> as it's just not necessarily true in the ring of algebraic integers.
>
> Why not necessarily true?
>
> Because the exclusion does NOT apply if 1-x is a unit in the ring of
> algebraic integers.
>
> So if it is a unit then the series can converge within the ring, but
> otherwise the exclusionary rule--the definition of algebraic integers
> as roots of monic polynomials--prevents.
>
Yes. We all agree on this. Why is it bad?
The series
S = 1 + x + x^2 + x^3 +...
converges (for abs(x) < 1) in the ring of real numbers. But that
ring is a field, and I suspect for your purposes you are not
interested in rings which are fields.
If x is an *object* and abs(x) < 1, does the series converge in
your ring of objects? I mean, if nonconvergence in the ring of
algebraic integers is a problem for some reason, does the ring of
objects solve that problem?
Marcus.
> Now I feel confident there are posters who will want to dispute me on
> the ability to use convergent infinite series in the ring of algebraic
> integers, but I want more than namecalling, or other childish rants,
> and I want more than someone saying they read it different in some
> number theory text.
>
[political platform deleted]
>
> James Harris
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